How to add image texture material to all parts of maze?
There are a few issues.
They are easy to repair, but will take some time.



1) Your original maze contains overlayed geometry.
That's a big problem in 3D modelling.
You will need to clean it up. It's technically very easy, but might take some time.

Clean up overlayed geometry and connect all vertices like this.


2) Keep quads whenever possible.
I cannot overstate this - this maze is begging to be made of quads.
Quads will allow you to use loopcuts and other useful features to speed up your workflow.

3) Keep scale to 1.
After transforms, go to Object->Apply-Scale.
This will ensure that UV mapping works properly.  

Your current maze mesh has a messed up scale. 


4) Use Solidify modifier to adjust the height of your maze.
You just need to Apply the modifier before subdivision or UV mapping.  

5) Use Smart UV project for UV unwrapping.
This function is well suited for complex geometric shapes such as your maze.

Here is a page from Blender Manual explaining Smart UV project.



Good luck.
If you have any questions, let me know.

Here is your edited Blendfile.
I separated the bottom part of your maze and connected a few overlayed faces.